L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. Journal of Symbolic Logic, to appear.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that certain properties of finite structures are not expressible in first order logic, and it seems that this was Gaifman s main motivation. More recently, Libkin and others considered this technique of proving inexpressibility results using locality in a complexity theoretic context (see, e.g. [5, 14, 13, 15]) A completely different application of Gaifman s theorem has been proposed in [11] It can be used to evaluate first order sentences in certain finite structures quite efficiently. In general, it takes time n (l) to decide whether a structure of size n satisfies a firstorder sentence of size ....

L. Hella, L. Libkin, and Y. Nurmonen. Notions of locality and their logical characterizations over finite models. Journal of Symbolic Logic, 64:1751--1773, 1999.

....that certain properties of finite structures are not expressible in first order logic, and it seems that this was Gaifman s main motivation. More recently, Libkin and others considered this technique of proving inexpressibility results using locality in a complexity theoretic context (see, e.g. [5, 15, 14, 16]) A completely different application of Gaifman s theorem has been proposed in [11] It can be used to evaluate first order sentences in certain finite structures quite efficiently. In general, it takes time n (l) to decide whether a structure of size n satisfies a first order sentence of size ....

L. Hella, L. Libkin, and Y. Nurmonen. Notions of locality and their logical characterizations over finite models. Journal of Symbolic Logic, 64:1751--1773, 1999.

....expressed by a first order formula. For example, to decide whether there is a path between two vertices of a graph it clearly does not suffice to look at small neighborhoods of these vertices. Hence by locality, s t connectivity is not expressible in first order logic. Recently, Libkin and others [2, 9 13] systematically started to explore locality as a tool for proving inexpressibility results. The ultimate goal of this line of research would be to separate complexity classes, in particular, to separate the class TC 0 from LOGSPACE. 1 A result of Hella and the first author (unpublished) showing ....

L. Hella, L. Libkin, and Y. Nurmonen. Notions of locality and their logical characterizations over finite models, 1998. To appear in Journal of Symbolic Logic.

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L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. Journal of Symbolic Logic, to appear.

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L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. J. Symb. Logic, to appear.

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L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. J. Symb. Logic, 64 (1999), 1751--1773.

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L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over nite models. J. Symbolic Logic 64(4): 1751-1773 (1999).

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L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. J. Symbolic Logic 64(4): 1751--1773 (1999).

....Definition 4.5. see [19; 14; 30; 21] A formula ( x) on pure two sorted structures is called Hanf local if there exist a number d 0 such that for all finite one sorted structures A and B, A; a) d (B; b) implies A j= a) iff B b) The definition for open formulae is from [21]; most previous papers [19; 14; 30; 37] considered its restriction to sentences. It is known [14] that A d B implies A r B for r d. It is also known that every (one sorted) first order sentence Phi is Hanf local and d can be taken to be 3 [14] This was generalized to various counting logics ....

....= n, where TrCl(A) is the transitive closure of A. Thus, the transitive closure query cannot be expressible in any logic that has the BNDP. The relationship between the notions of locality we introduced is the following, when one deals with one sorted finite structures: Proposition 4.8. a) see [21]) Every Hanf local formula is Gaifman local. b) see [11] Every query defined by a Gaifman local formula has the BNDP. 2 These results are not affected by the transfer to pure two sorted structures. 4.3 Locality of In [37] it was proved that the extension of first order logic by all unary ....

L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. Journal of Symbolic Logic, 64 (1999), 1751-1773.

....evidence that SQL cannot express recursive queries For that purpose, we shall use the locality of queries. Locality was the basis of a number of tools for proving expressivity bounds of rst order logic [15,13,11] and it was recently studied on its own and applied to more expressive logics [17,23]. The general idea of this notion is that a query can only look at a small portion of its input. If the input is a graph, small means a neighborhood of a xed radius. For example, Fig. 1 shows that reachability is not local: just take a graph like the one shown in the picture so that there ....

....Could it be that order independent queries in sql N and sql N are the same Of course, such a result would imply that TC is properly contained in DLOGSPACE, and several papers suggested this approach towards separating complexity classes. Unfortunately, it does not work, as shown in [17]: Proposition 3 There exist order independent non local queries expressible in N . Thus, there are order independent sql N queries not expressible in sql N . Proof. It was shown in [17] that, on the graph of an n element successor relation with an extra predicate P interpreted as the rst ....

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L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over nite models. Journal of Symbolic Logic, 64 (1999), 17511773. 30

....with constants in terpreted as 5. That is, for Nr(5) and Nr( to be isomorphic (written N, 5) N, we must have an isomorphism h such that h(5) 5. Suppose we have an ra ary query, that is, a mapping Q that associates with each structure 1 a subset of A TM. We say that Q is Gaifman local [4] if there ex ists a number r such that for any structure 1 and any 5,e A TM, Intuitively, if Q is a local query and two tuples look alike in some neighborhood, then they are indistinguishable by Q. Gaifman s locality theorem [2] implies that every FO definable query is Gaifman local. More ....

....there ex ists a number r such that for any structure 1 and any 5,e A TM, Intuitively, if Q is a local query and two tuples look alike in some neighborhood, then they are indistinguishable by Q. Gaifman s locality theorem [2] implies that every FO definable query is Gaifman local. More recently [4, 5] it was shown that Gaifman locality extends to logics with very powerful counting mechanisms. While Gaifman locality is very useful for proving expressivity bounds, its limitations are well understood [4] in par ticular, it does not apply to the extension of FO with counting quantifiers and a ....

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L. Hella, L. Libkin, and J. Nurmonen. Notions of locality and their logical characterizations over finite models. Journal of Symbolic Logic, 64:1751-1773, 1999.

.... We write a D r b if N D r ( a) and N D r ( b) are isomorphic; that is, if there exists a one to one map h : S D r ( a) S D r ( b) such that h( a) b and t 2 R i i h( t) 2 R i , for every i l and a tuple t of elements of S D r ( a) De nition 1 (cf. [11, 17]) An n ary query Q is called local if there exists a number r 0 such that, for any database D 2 Inst(SC in ) and any a; b 2 adom(D) n , a D r b implies a 2 Q( A) i b 2 Q( A) The minimum such r is called the locality rank of Q, and is denoted by lr(Q) A language is ....

L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over nite models. Journal of Symbolic Logic, 64 (1999), 1751-1773.

....Definition 4.5. see [19; 14; 30; 21] A formula ( x) on pure two sorted structures is called Hanf local if there exist a number d 0 such that for all finite one sorted structures A and B, A; a) d (B; b) implies A j= a) iff B j= b) The definition for open formulae is from [21]; most previous papers [19; 14; 30; 37] considered its 14 Delta L. Hella, L. Libkin, J. Nurmonen, L. Wong restriction to sentences. It is known [14] that A d B implies A r B for r d. It is also known that every (one sorted) first order sentence Phi is Hanf local and d can be taken to be 3 ....

....of A. Thus, the transitive closure query cannot be expressible in any logic that has the BNDP. The relationship between the notions of locality we introduced is the following, when one deals Logics with Aggregate Operators Delta 15 with one sorted finite structures: Proposition 4.8. a) see [21]) Every Hanf local formula is Gaifman local. b) see [11] Every query defined by a Gaifman local formula has the BNDP. 2 These results are not affected by the transfer to pure two sorted structures. 4.3 Locality of LC In [37] it was proved that the extension of first order logic by all unary ....

L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. Journal of Symbolic Logic, 64 (1999), 1751-1773.

....that HQ = f . Thus, dropping a tiny portion of linear order (e.g. log log : log n elements) accounts for the increase in hardness from constant to arbitrary one FO(C) also admits this kind of dichotomy, as there exists a colored graph query Q definable in FO(C) such that HQ (n) log n [13]. We thus obtain: Corollary 4 There are problems in uniform TC 0 that cannot be expressed in FO(C) g . QED Moreover, it is known that there are uniform AC 0 (equivalently, first order logic with the BIT predict and a linear order, FO(BIT) queries that violate the BNDP [6,11] Hence, we ....

.... that (A n ; P ) j= fi(a; b) Since there is no path between d i and c i for every i, we have (A n ; P ) j= fi(a; b) Thus, we must show how to express fl in L 1 (C) in fact, one can express it in FO(C) To express fl, we follow the proof of the failure of the BNDP for FO(C) given in [13]. Let P 1 ff = U 1 P ff . Since jP 1 ff j jU 1 j = M n log(jP ff j) subsets of P 1 ff can be coded by the elements of P ff : a set S P 1 ff is coded by c S 2 P ff such that fx j BIT(m 1 ; m 2 ) where m 1 = jfy j y xgj ; m 2 = jfy j y c S gj g = S With this coding, we can ....

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L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. Journal of Symbolic Logic, 64 (1999), 1751-- 1773.

....of these tuples are isomorphic; again d is determined by k. It was shown that Hanf s theorem is strictly stronger than Gaifman s, and that both apply to a variety of logics that extend FO with counting mechanisms and limited in nitary connectives [Grohe and Schwentick 2000; Hella et al. 1999a; Hella et al. 1999b; Libkin 2000; Nurmonen 1996] These results found applications in descriptive complexity and database theory. Since the complexity class TC 0 (with the appropriate notion of uniformity) can be captured by FO with counting quanti ers [Barrington et al. 1990] locality can be used to prove lower bounds for ....

....of this logic, which is one sorted, and uses arbitrary unary generalized quanti ers [Hella 1996; Hella et al. 1999a] however, expressing counting properties with unary quanti ers is often quite awkward, and thus we chose to use a two sorted version with counting terms here. Fact 2.7. See [Hella et al. 1999b; Libkin 2000]. Queries expressed by L 1 (C) formulae without free variables of the second sort are Hanf local and Gaifman local. Gaifman locality of L 1 (C) was proved by a simple direct argument in [Libkin 2000] Hanf locality was shown in [Hella et al. 1999b] using bijective EhrenfeuctFra ss e ....

Hella, L., Libkin, L. and Nurmonen, J. 1999a. Notions of locality and their logical characterizations over nite models. J. Symb. Logic 64, 4, 1751-1773.

....evidence that SQL cannot express recursive queries For that purpose, we shall use the locality of queries. Locality was the basis of a number of tools for proving expressivity bounds of first order logic [15, 13, 11] and it was recently studied on its own and applied to more expressive logics [17, 22]. b : oe : r r : a oe Upsilon Sigma Pi Xi Upsilon Sigma Pi Xi Fig. 1. A local formula cannot distinguish (a; b) from (b; a) The general idea of this notion is that a query can only look at a small portion of its ....

....Could it be that order independent queries in sql N and sql N are the same Of course, such a result would imply that TC 0 is properly contained in DLOGSPACE, and several papers suggested this approach towards separating complexity classes. Unfortunately, it does not work, as shown in [17]: Proposition 3. There exist order independent non local queries expressible in sql N . Thus, there are order independent sql N queries not expressible in sql N . Proof sketch. On the graph of an n element successor relation with an extra predicate P interpreted as the first blog 2 nc ....

L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. J. Symb. Logic, 64 (1999), 1751-1773.

....is determined only by a formula defining the query, and the maximum degree of the input graph, but not the size of the graph [24] As there are several ways to define locality, we want to establish the strongest property. The relationship between various notions of locality was investigated in [14, 21], and it was shown that the one based on Hanf s theorem implies the one based on Gaifman s theorem, which in turn implies the property stated in the previous paragraph. Thus, our goal is to show (precise definition will be given a bit later in this section) Expressiveness of L aggr (All; All) ....

....a bijection f : A B such that ntp A d ( ac) ntp B d ( bf(c) for every c 2 A. Hanf locality has been previously defined only for finite one sorted structures. In the following we make a natural extension of its definition to the case of pure two sorted structures. Definition 4. 3 (see [12, 8, 21, 14]) A formula ( x) on pure two sorted structures is called Hanf local if there exist a number d 0 such that for all finite one sorted structures A and B, A; a) d (B; b) implies A j= a) iff B j= b) The definition for open formulae is from [14] most previous papers [12, 8, 21, 28] ....

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L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. J. Symb. Logic, to appear.

....for Hanf locality uses a logic L 1 (C) introduced in [Li00] This logic subsumes a number of counting extensions of FO (such as FO with counting quantifiers [IL90] FO with unary generalized quantifiers [He96; KV95] FO with unary counters [BK97] and is quite easy to deal with. A result in [HLN99] states that Hanf local properties on structures of bounded valence are precisely those definable in L 1 (C) The question naturally arises whether this continues to hold for arbitrary finite structures. We show in this paper that this is not the case. We do so by first finding a simple direct ....

....that are only allowed to range over fixed radius neighborhoods of free first order variables. We will also show that this amounts to adding arbitrarily powerful computations to L 1 (C) as long as they are bound to some neighborhoods. For Gaifman locality, a characterization theorem in [HLN99] stated that it is equivalent, over structures of bounded valence, to first order definition by cases. That is, there are m 0 classes of structures and m FO formulae i such that over the ith class, the given property is described by i . Again, this falls short of a general characterization. ....

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L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. J. Symb. Logic, 64 (1999), 1751--1773.

....for every c 2 A. Definition 2.4 (Hanf locality) a) see [15; 10; 24] A sentence Phi is called Hanf local if there exist a number d 0 such that A and B agree on Phi whenever A d B. The minimum d for which this holds is called Hanf locality rank of Phi, and is denoted by hlr( Phi) b) see [17]) A formula ( x) is called Hanf local if there exist a number d 0 such that for any A; B, and a; b over A and B, resp. of the same length as x, A; a) d (B; b) implies A j= a) iff B j= b) It can be seen [10] that A d B implies A r B for r d; in particular, if A d B, then j A ....

....d (B; b) implies A j= a) iff B j= b) It can be seen [10] that A d B implies A r B for r d; in particular, if A d B, then j A j=jB j. Fact 2.5. a) see [10] If A 3 nB, then A jn B. In particular, A and B agree on all FO sentences of quantifier rank up to n. b) see [27] bound from [17]) Let n 0. Then A (3 n Gamma1 Gamma1) 2 B implies A j bij n B. c) see [17; 24] Every Hanf local formula (without free second sort variables, if one deals with a two sorted logic) is Gaifman local. 2 Next, we review results on outputs of local queries. With each formula (x 1 ; xn ) ....

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L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. Journal of Symbolic Logic, 64 (1999), 1751--1773.

....is shown for FO(Qu ) More bounds were obtained in [22] which used the results of [26] to prove an analog of Gaifman s locality theorem [11] for those logics. Currently, most bounds for extensions of FO with various counting quantifiers can be derived from its local properties, as shown in [17, 22, 26]; exceptions include the bound of [5] a result in [4] on counting the sizes of equivalence classes, and the hierarchy result in [14] Locality of a logic gives us a general statement that it lacks a recursion mechanism, much in the same way as 0 1 laws tell us that a logic cannot express ....

....us that a logic cannot express nontrivial counting properties. One way in which locality theorems are applied is the following. First, a form of locality based on Hanf s condition (see [10, 15] is shown for a logic; this form is closely tied to a gamecharacterization of the logic. Then results of [17, 22] show that the logic also satisfies Gaifman s locality condition [11] and the bounded degree property [24] which are much easier to apply to prove expressivity bounds. However, no direct proofs of those conditions have been given so far for any of the extensions of FO. The basic idea of locality ....

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L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. Manuscript, 1997.

....that HQ = f . Thus, dropping a tiny portion of linear order (e.g. log log : log n elements) accounts for the increase in hardness from constant to arbitrary one FO(C) also admits this kind of dichotomy, as there exists a colored graph query Q definable in FO(C) such that HQ (n) log n [17]. In particular, there are problems in uniform TC 0 that cannot be expressed in FO(C) g . Moreover, it is known that there are uniform AC 0 (that is, FO(BIT) queries that violate the BNDP ( 12] see also [6] Hence, we obtain: Corollary 3 AC 0 6 (L 1 (C) g ) w . 2 Corollary 3 ....

.... fl can be expressed follows from two observations: first, there are sufficiently many successor relations in E for one of them to be totally contained in P 1 , and second, on that successor relation, one can use the order part of P to code monadic second order using counting, as it was done in [17]. See [25] for details. 2 Proposition 1 provides the first nontrivial example that separates the notion of locality and the BNDP. Now one needs a different technique to prove Theorem 1. We introduce this technique in two steps. In the next section, we consider two ways of weakening the notion of ....

L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. J. Symb. Logic, to appear.

....extended by all unary generalized quantifiers [38] for the case of finite structures. Proofs of applicability of Hanf s technique typically are not very difficult [17, 15, 38, 40] We will see some examples in Section 4. The above results have motivated a study of general notions of locality [32, 25]. We review this line of work in Section 5. We show that Gaifman s theorem gives rise to two general notions, 2 one for sentences and one for open formulas. We formulate an abstract notion of locality that captures Hanf s condition, and study the relationship between the notions of locality. We ....

....rank of each formula is at most k. The union of all these logics L1 (Q u ) k over all natural numbers k is denoted by L1 (Q u ) that is, the depth of nesting of quantifiers in each formula is finite) Methods used in [22] give us the following result (a proof can be found in [25]) Theorem 3.6 (see [22, 25] Let A;B 2 STRUCT[oe] Then A j n bij B if and only if A and B agree on all L1 (Q u ) sentences of quantifier rank up to n. 2 Corollary 3.7 A class C STRUCT[oe] is not definable in L1 (Q u ) if and only if for every n there are A 2 C and B 62 C such that ....

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L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over finite models. Unpublished manuscript, 1997.

....that is determined only by a formula de ning the query, and the maximum degree of the input graph, but not the size of the graph [24] As there are several ways to de ne locality, we want to establish the strongest property. The relationship between various notions of locality was investigated in [14, 21], and it was shown that the one based on Hanf s theorem implies the one based on Gaifman s theorem, which in turn implies the property stated in the previous paragraph. Thus, our goal is to show (precise de nition will be given a bit later in this section) Expressiveness of L aggr (All; All) ....

....is a bijection f : A B such that ntp A d ( ac) ntp B d ( bf(c) for every c 2 A. Hanf locality has been previously de ned only for nite one sorted structures. In the following we make a natural extension of its de nition to the case of pure two sorted structures. De nition 4. 3 (see [12, 8, 21, 14]) A formula ( x) on pure two sorted structures is called Hanf local if there exist a number d 0 such that for all nite one sorted structures A and B, A; a) d (B; b) implies A j= a) i B j= b) The de nition for open formulae is from [14] most previous papers [12, 8, 21, ....

[Article contains additional citation context not shown here]

L. Hella, L. Libkin and J. Nurmonen. Notions of locality and their logical characterizations over nite models. J. Symb. Logic, to appear.

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L. Hella, L. Libkin, and J. Nurmonen. Notions of locality and their logical characterizations over nite models. Journal of symbolic logic 64, pages 17511773., 1999.

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L. Hella, L. Libkin, and Y. Nurmonen. Notions of locality and their logical characterizations over nite models, 1998. To appear in Journal of Symbolic Logic.

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